8 cosθ-1=0 to the nearest 0.1°

The regular-size granola bar is 1 1/3 inches longer than the snack-size granola bar.

The regular-size granola bar has a volume of 4 cubic inches, while the snack-size bar has a volume of 3 cubic inches.

Since both bars have the same width and height, we can use the formula for the volume of a rectangular prism to find the length of each bar:

Regular-size bar: V = lwh = 4 cubic inches, w = 1 inch, h = 3/4 inch

l = V/wh = 4/(1 × 3/4) = 16/3 inches

Snack-size bar: V = lwh = 3 cubic inches, w = 1 inch, h = 3/4 inch

l = V/wh = 3/(1 × 3/4) = 4 inches

Therefore, the regular-size granola bar is (16/3 - 4) = 4/3 inches longer than the snack-size granola bar.

This can also be written as the mixed number 1 1/3 inches.

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It would cost [tex]$8,100[/tex] to rent the safety deposit box for 15 years at a rate of [tex]$45[/tex] per month.

The total cost of renting a safety deposit box for 15 years, we need to first determine the total number of months in 15 years.

Since there are 12 months in a year, the total number of months in 15 years is:

15 years x 12 months/year = 180 months

So, if the safety deposit box costs [tex]$45[/tex] per month to rent, then the total cost of renting the box for 15 years would be:

180 months x [tex]$45[/tex]/month = [tex]$8,100[/tex]

We must first ascertain the entire number of months in 15 years in order to compute the total cost of renting a safety deposit box for 15 years.

Since there are 12 months in a year, there are a total of 180 months in 15 years: 15 years multiplied by 12 months per year.

In this case, if the monthly rental fee for the safety deposit box is [tex]$45[/tex] and the rental period is 15 years, the total cost would be calculated

180 months x [tex]$45[/tex]/month = [tex]$8,100[/tex]

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The Second Linearly Independent solution of the differential equation is:

y = c₁x⁷ + c₂(Ax¹⁰ + Bx⁻¹¹)

When solving a second-order linear differential equation of the form

x²y'' - 42y = 0, it is important to find two linearly independent solutions to fully describe the general solution. The first solution is given as y₁=x⁷.

To find the second linearly independent solution, we can use the method of reduction of order.

Let y₂ = u(x)y₁(x), where u(x) is a function to be determined.

Then we have y₂' = u(x)y₁'(x) + u'(x)y₁(x) and y₂'' = u(x)y₁''(x) + 2u'(x)y₁'(x) + u''(x)y₁(x).

Substituting y₂ and its derivatives into the original differential equation, we have:

x²(u(x)y₁''(x) + 2u'(x)y₁'(x) + u''(x)y₁(x)) - 42u(x)y₁(x) = 0

Dividing by x²y₁(x), we get:

u''(x)/u(x) + 2/x[u'(x)/u(x)] - 42/x² = 0

Let v(x) = u'(x)/u(x), then v'(x) = u''(x)/u(x) - (u'(x))²/(u(x))². Substituting v(x) into the above equation, we have:

v'(x) + 2/xv(x) - 42/x² = 0

This is now a first-order linear differential equation that can be solved using an integrating factor. Letting mu(x) = x², we have:

(x²v(x))' = 42

Solving for v(x), we get:

v(x) = 21/x + C/x²

where C is an arbitrary constant. Substituting back to u(x), we get:

u(x) = Ax³ + Bx⁻⁻¹⁸

where A and B are constants. Therefore, the second linearly independent solution is

y₂ = (Ax³ + Bx⁻¹⁸)x⁷ = Ax¹⁰ + Bx⁻¹¹

Hence, the general solution of the differential equation is:

y = c₁x⁷ + c₂(Ax¹⁰ + Bx⁻¹¹)

where c₁ and c₂ are arbitrary constants

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The complete statement should be If 3m = 7n, then m/n = 7/3. The proportion was obtained by solving for m, and then using the converse of the cross products property. Option B

The converse of the cross products property states that if two ratios are equal, then the product of the means is equal to the product of the extremes.

To justify the answer, you solve for m/n in the original equation by dividing each side by 3n, which gives you m/n = 7/3.

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Answer:

To determine the sample size required to estimate the proportion of consumers who have seen the company's advertisement in the newspaper, we can use the formula for sample size calculation for proportions. The formula is as follows:

n = (Z^2 * p * q) / E^2

Where:

n = required sample size

Z = Z-score corresponding to the desired confidence level (in this case, 98% confidence corresponds to a Z-score of approximately 2.33)

p = estimated proportion (0.65)

q = 1 - p (the complement of the estimated proportion)

E = maximum error tolerance (+/- 0.05)

Let's plug in the values and calculate the sample size:

n = (2.33^2 * 0.65 * 0.35) / (0.05^2)

n = 339.28

Rounding up to the nearest whole number, the required sample size is 340.

For calculating the percentage of discount, we can use the formula:

Percentage of discount = (Discount amount / Original price) * 100

We have given that the discounted price is Birr 51,000 and the discount amount is Birr 9,000,we need to find the original price.

Original price = Discounted price + Discount amount

Original price = 51,000 + 9,000 = 60,000 Birr

Now, we can calculate the percentage of discount:

Percentage of discount = (9,000 / 60,000) * 100 = 15%

Hence, the percentage of discount for the four-door model is 15%.

The correct answer is [tex]\dfrac{16\pi }{9}[/tex]

The area of a shaded region is [tex]\sf \dfrac{n\pi r^2}{360}[/tex]

So, applying the formula [tex]\sf \dfrac{n\pi r^2}{360}[/tex]

[tex]\sf =\dfrac{160\pi r^2}{360}=\dfrac{160\pi (2^2)}{360} =\dfrac{160\times\pi \times4}{360} =\dfrac{16\pi }{9}[/tex]

Hence the area of shaded region is [tex]\dfrac{16\pi }{9}[/tex]

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Answer:

1/12 gallons

Step-by-step explanation:

We can determine how much more hot chocolate than apple cider die Joe serve by subtracting 1/12 from 1/6.

Step 1:  Since 1/12 has the bigger denominator and 6 * 2 = 12, let's give 1/6 the same denominator as 1/12 by multiplying the entire fraction by 2/2

(1/6) * (2/2) = 2/12

Step 2:  Now we can subtract 1/12 from 2/12:

2/12 - 1/12 = 1/12

Thus, Joel served 1/12 more gallons of hot chocolate than apple cider.

Answer: 187 apples

Step-by-step explanation:

Substitute "year 5" into the equation. y = 15.2(5) +111.

y = 76 + 111

y = 187

We have that Linda's net and expression incorrect

With the Expression of the surface area of attic

45 (40 + 25 + 25) + 1/2 (40 x 15)

Hence, Surface area is,

X = 4050 + 600

X = 4650 feet²

Hence, The answer is No, Because Her expression for the triangles are incorrect;

1/2(40 x 15)

Instead of,

1/2 x 40 x 15

Hence, Linda's net and expression incorrect,

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Linda is adding padding to all of the surfaces inside her attic for extra warmth in the winter.

She needs to find the approximate surface area of the attic, including the walls, floor, and

ceiling. The attic is in the shape of a triangular prism. Linda draws the net and writes

the expression below to represent the surface area of the attic. Are Linda's net and

expression correct?

14. The measure of angle ABC is 90⁰.

15. The measure of angle CED is 43⁰.

16. The measure of angle BDE is 47.5⁰.

17. The measure of angle CBD is 43⁰.

18. The measure of arc AD is 94⁰.

19. The measure of angle BCE is 47.5⁰.

20. The measure of angle ABD is 47⁰.

21.  The measure of arc ABC is 180⁰.

The measure of the missing angles is calculated by applying intersecting chord theorem, which states that the angle at tangent is half of the arc angle of the two intersecting chords.

The measure of angle ABC is calculated as follows;

m∠ABC = 90⁰ (since line AC is the diameter)

The measure of angle CED is calculated as follows;

m∠CED = ¹/₂ (arc CD) (interior angle of intersecting secants)

m∠CED = ¹/₂ (86⁰) = 43⁰

The measure of angle BDE is calculated as follows;

m∠BDE= ¹/₂ (arc BE) (interior angle of intersecting secants)

m∠BDE = ¹/₂ (95⁰) = 47.5⁰

The measure of angle CBD is calculated as follows;

m∠CBD = ¹/₂ (arc CD) (interior angle of intersecting secants)

m∠CBD = ¹/₂ (86⁰) = 43⁰

The measure of arc AD is calculated as follows;

arc ABC = 180 (sum of angles in a semi circle)

arc ADC = 180 (sum of angles in a semi circle)

arc ADC = arc AD + arc CD

180 = AD + 86

AD = 94⁰

The measure of angle BCE is calculated as follows;

m∠BCE= ¹/₂ (arc BE) (interior angle of intersecting secants)

m∠BCE = ¹/₂ (95⁰) = 47.5⁰

The measure of angle ABD is calculated as follows;

m∠ABD = ¹/₂ (arc AD) (interior angle of intersecting secants)

m∠ABD = ¹/₂ (94⁰) = 47⁰

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Answer:9/41

SOH = opposite over hypotenuse

Step-by-step explanation:

a) The rate of slower train is 110 km/hr

b) The speed of faster train is 112 km/hr

Given data ,

When two objects are moving towards each other, their combined rate is equal to the sum of their individual speeds.

So, the rate of the slower train as x km/hr. Since the faster train is traveling 12 km/hr faster, its rate will be ( x + 12 ) km/hr.

The combined rate of the two trains is x + (x + 12) = 2x + 12 km/hr.

We know that the trains travel a total distance of 696 km and meet in 3 hours. Using the formula distance = speed × time, we can set up the following equation:

(2x + 12) × 3 = 696

Simplifying the equation:

6x + 36 = 696

Subtracting 36 from both sides:

6x = 660

Dividing both sides by 6:

x = 110

Hence , the rate of the slower train is 110 km/hr, and the rate of the faster train is 110 + 12 = 122 km/hr.

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The standard form (SF), center (C) and radius (R) are given as follow: (13) SF:  (x + 4)² + (y - 3)² = 50, Center: (-4, 3), R: √50 (14) SF: (x - 2)² + (y - 6)² = 169, C: (2, 6), R: 13 (15) SF: (x + 7)² + (y + 5)² = 1, C: (-7, -5), R: 1 (16) SF: (x - 8)² + y² = 225, C: (8,0), R: 15 (17) SF: (x - 12)² + (y - 2)² = 63, C: (12, 2), R: √63 (18) SF: (x - 5)² + (y + 4)² = 100 , C: (5, -4), R: 10

The general form of a circle is given as:

       (x - h)² + (y - k)² = r²

where:

        (h, k) represents the center of the circle

       r represents the radius.

Now we can use the above information to solve the following questions:

13. x² + y² + 8x - 6y - 25 = 0

Rearranging the equation:

x² + 8x + y² - 6y = 25

Completing the square for x terms:

(x² + 8x + 16) + y² - 6y = 25 + 16

Simplifying:

(x + 4)² + (y² - 6y) = 41

(x + 4)² + (y² - 6y + 9) = 41 + 9

(x + 4)² + (y - 3)² = 50

Center: (-4, 3)

Radius: √50

14. x² + y² - 4x - 12y - 129 = 0

Rearranging the equation:

x² - 4x + y² - 12y = 129

Completing the square for x terms:

(x² - 4x + 4) + y² - 12y = 129 + 4

Simplifying:

(x - 2)² + (y² - 12y) = 133

(x - 2)² + (y² - 12y + 36) = 133 + 36

(x - 2)² + (y - 6)² = 169

Center: (2, 6)

Radius: 13

15. x² + y² + 14x + 10y + 73 = 0

Rearranging the equation:

x² + 14x + y² + 10y = -73

Completing the square for x terms:

(x² + 14x + 49) + y² + 10y = -73 + 49

Simplifying:

(x + 7)² + (y² + 10y) = -24

(x + 7)² + (y² + 10y + 25) = -24 + 25

(x + 7)² + (y + 5)² = 1

Center: (-7, -5)

Radius: 1

16. x² + y² - 16x - 161 = 0

Rearranging the equation:

x² - 16x + y² = 161

Completing the square for x terms:

(x² - 16x + 64) + y² = 161 + 64

Simplifying:

(x - 8)² + y² = 225

Center: (8, 0)

Radius: 15

17. x² + y² = 24x + 4y - 85

Rearranging the equation:

x² - 24x + y² - 4y = -85

Completing the square for x and y terms:

(x² - 24x + 144) + (y² - 4y + 4) = -85 + 144 + 4

Simplifying:

(x - 12)² + (y - 2)² = 63

Center: (12, 2)

Radius: √63

18. x² + y² - 9x + 2y = x - 6y + 59

Rearranging the equation:

x² - 9x - x + y² + 2y + 6y = 59

Combining like terms:

x² - 10x + y² + 8y = 59

Completing the square for x and y terms:

(x² - 10x + 25) + (y² + 8y + 16) = 59 + 25 + 16

Simplifying:

(x - 5)² + (y + 4)² = 100

Center: (5, -4)

Radius: 10

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The value of a in the triangle, given that in the triangle ABC, b = 2, c = 6, and A = 35° is 7.7 (option A)

We can obtain the value of a in the triangle ABC as shown in the attached photo by using the cosine rule as illustrated below:

Cosine rule states as follow:

a² = b² + c² + 2bc Cos A

Inputting the given parameters, we can obtain the value of a as follow:

a² = 2² + 6² + (2 × 2 × 6 × Cos 35)

Clear the bracket

a² = 4 + 36 + 19.66

a² = 59.66

Take the square root of both sides

a = √59.66

a = 7.7

Thus, we can conclude from the above calculation that the value of a is 7.7 (option A)

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The required angles (corresponding, vertical and alternate) in relation to the Parallel lines are attached accordingly.

Parallel lines are coplanar infinite straight lines that do not cross at any point in geometry. Parallel planes are planes that never intersect in the same three-dimensional space.

When two parallel lines cross by any other line (i.e. the transversal), corresponding angles are generated in matching corners or corresponding corners with the transversal.

When two parallel lines are sliced by a transversal, the resulting alternate exterior angles are congruent, according to the Alternate Exterior Angles Theorem.

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Answer: x=25 degrees

Step-by-step explanation:

25 degrees (blue) and x degrees (orange) are vertical angles. Vertical angles always equal each other.

(80 degrees {green} is just there to throw you off)

The value of y from the given right angled triangle is 9.2 cm.

Given that, a right angled triangle has 8.2 cm, y cm and 12.3 cm.

Let, hypotenuse = 12.3 cm, perpendicular = 8.2 cm and base = y cm.

By using Pythagoras theorem, we get

8.2²+y²=12.3²

67.24+y²=151.29

y²=151.29-67.24

y²=84.05

y=9.2 cm

Therefore, the value of y from the given right angled triangle is 9.2 cm.

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Given statement solution is :- Outputs and values g(2/3) = 2/3.

To summarize:

g(0) = 0

g(-1) = undefined

g(2) = undefined

g(2/3) = 2/3

To find the values of g(x) for the given inputs, we substitute each input into the function g(x) = x / √([tex]1 - x^2[/tex]). Let's calculate the values:

g(0):

Substitute x = 0 into the function:

g(0) = 0 / √([tex]1 - 0^2[/tex])

= 0 / √(1 - 0)

= 0 / √1

= 0

Therefore, g(0) = 0.

g(-1):

Substitute x = -1 into the function:

g(-1) = (-1) / √(1 - [tex](-1)^2[/tex])

= (-1) / √(1 - 1)

= (-1) / √0

Since the square root of 0 is undefined, g(-1) is undefined.

g(2):

Substitute x = 2 into the function:

g(2) = 2 / √([tex]1 - 2^2[/tex])

= 2 / √(1 - 4)

= 2 / √(-3)

Since the square root of a negative number is undefined in the real number system, g(2) is undefined.

g(2/3):

Substitute x = 2/3 into the function:

g(2/3) = (2/3) / √(1 - [tex](2/3)^2[/tex])

= (2/3) / √(1 - 4/9)

= (2/3) / √(5/9)

= (2/3) / (√5/√9)

= (2/3) / (√5/3)

= (2/3) * (3/√5)

= 2√5 / 3√5

= 2/3

Therefore, outputs and values g(2/3) = 2/3.

To summarize:

g(0) = 0

g(-1) = undefined

g(2) = undefined

g(2/3) = 2/3

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Answer:

[tex] \sin(x) + \sin(2x) = 0[/tex]

[tex] \sin(x) + 2 \sin(x) \cos(x) = 0[/tex]

[tex]( \sin(x) )(1 + 2 \cos(x) ) = 0[/tex]

sin(x) = 0 or 1 + 2cos(x) = 0

2cos(x) = -1

cos(x) = -1/2

x = kπ or x = 2π/3 + 2kπ or

x = 4π/3 + 2kπ

(k is an integer)

The value of R and θ is 6.18 and 53.13 degrees, under the condition that a circle C has center at the origin and radius 5 .

In order to  evaluate the equation of the circle K. The diameter of K has endpoints at the origin and (0,15). Then, the center of K is at (0,7.5) and its radius is 7.5. Therefore, the evaluated equation of K is
x² + (y-7.5)² = 56.25.
The equation of circle C is x² + y² = 25.
The two circles intersect at two points. Now we have to evaluate the coordinates of these points.
Staging y = 5 - x² in the equation of K,
we get
x² + (5-x²-7.5)² = 56.25.
Applying simplification on this equation
x⁴ - 10x² + 31.25 = 0.
Calculating this quadratic equation gives us
x² = 5 ± √(10)/2.
If P lies in the first quadrant,
we choose x² = 5 + √(10)/2 and y = √(25-x²) to get P in Cartesian coordinates.
Converting P to polar coordinates gives us
r = √(x²+y²) and θ = arctan(y/x).
Staging x = √(5+√(10)/2) and y = √(25-x²) in these equations gives us
r ≈ 6.18 and θ ≈ 0.93 radians.
Using this formula to convert into degree
Rad × 180/π
= 0.93 × 180/π
≈ 53.13 degrees
Therefore, r ≈ 6.18 and θ ≈ 53.13 degrees.
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The solution to the function h(x) = 2x² - x - 1 is given by the graph attached which is a positive quadratic graph.

A quadratic graph represents a quadratic equation in the form:

   y = ax² + bx + c,

where a, b, and c are constants.

The graph of a quadratic equation is a curve called a parabola.

The general shape of a quadratic graph depends on the value of the coefficient "a." If "a" is positive, the graph opens upward, and if "a" is negative, the graph opens downward.

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The angle measure of LJK is 27 degrees

Following the triangle sum theorem, we have that the sum of the interior angles of a triangle is 180 degrees

Also, we need to know that the sum of angles on a straight line is 180, then, we have;

<JLK = 180 - <LKM = 180 - (8x - 19)

Then, substitute the value, we have that;

<JKL + < JLK + < KJL = 180

Then,

3x - 16 + (180 - (8x - 19)) + 2x + 15 = 180

expand the bracket, we have;

3x - 16 - 8x - 19 + 2x + 15 = 0

add the like terms

-3x + 18 = 0

collect the terms

-3x = -18

x = 6

Then, the angle LJK = 2x + 15 = 2(6) + 15 = 27 degrees

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Answer:

Step-by-step explanation:

Let's factor the expression w^2(w+8)-5(w+8):

First, we can see that (w+8) is a common factor in both terms of the expression.

So, we can factor out (w+8) from both terms:

(w+8)(w^2 - 5)

Now, the expression is factored as (w+8)(w^2 - 5).

The factored form of the polynomial [tex]x^4 - x^3 - 5x^2 - x - 6[/tex] is:

[tex](x - 2)(x^3 + x^2 - 3x - 7)[/tex]

We have,

To factor the polynomial [tex]x^4 - x^3 - 5x^2 - x - 6,[/tex]

we can look for its roots and express it as a product of linear factors.

First, let's check if there are any rational roots using the rational root theorem.

The possible rational roots can be found by taking the factors of the constant term (-6) and dividing them by the factors of the leading coefficient (1).

The factors of -6 are: ±1, ±2, ±3, ±6

The factors of 1 are: ±1

The possible rational roots are: ±1, ±2, ±3, ±6

By testing these values, we find that x = 2 is a root of the polynomial.

Using synthetic division, we can divide the polynomial by (x - 2) to find the quotient.

The quotient is [tex]x^3 + x^2 - 3x - 7.[/tex]

Now, we can continue factoring the quotient.

The polynomial x³ + x² - 3x - 7 does not have any rational roots.

We can try factoring it by grouping or using other factoring methods, but in this case, it does not factor nicely into linear factors.

Therefore,

The factored form of the polynomial [tex]x^4 - x^3 - 5x^2 - x - 6[/tex] is:

[tex](x - 2)(x^3 + x^2 - 3x - 7)[/tex]

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Answer:

8

Step-by-step explanation:

3x - 10 = x + 2

2x = 12

x = 6

3x - 10 = 3(6) - 10 = 8

The length of each leg of the triangle is 8.

ΔAOB will be neither isosceles nor a right angled triangle.

Given,

In Parallelogram ABCD diagonals bisect at the point O.

Parallelogram and its properties:

Hence from the above properties it is clear that the triangles formed will neither be isosceles nor a right angled triangle.

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No, 34.05 is not greater than 34.59.

When comparing two numbers, we look at their digits from left to right.

In this case, both numbers start with 34, which means they have the same value in the tens place.

However, when we move to the decimal part, 0.05 is smaller than 0.59.

Therefore, 34.05 is smaller than 34.59.

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The length of fencing needed around her triangular garden is 41 feet.

A given shape which has three sides and three measures of internal angles which add up to 180^o is said to be a triangle.

For Mary to build a fence around her triangular garden, the amount of fencing needed can be determined by adding the length of each side of the garden.

So that;

A + B + C = 180^o

A + 49 + 40 = 180

A = 180 - 89

   = 91

A = 91^o

Applying the sine rule, we have;

a/Sin A = b/Sin B = c/Sin C

a/Sin A = b/Sin B

a/Sin 91 = 13/ Sin 49

aSin 49 = 13*Sin 91

             = 12.998

a = 12.998/ 0.7547

  = 17.22

a = 17 ft

Also,

b/Sin B = c/Sin C

13/Sin 49 = c/ Sin 40

cSin 49 = 13*Sin 40

             = 8.3562

c = 8.3562/ 0.7547

  = 11.0722

c = 11 feet

Thus the amount of fencing required = 13 + 17 + 11

                                                    = 41

The fencing required is 41 feet length.

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The equation of the line is: x = 1 + 3t,, y = 10, z = 3 + 3t.

To determine the equation of the line passing through points M(1, 10, 3), N(4, 10, 6), and P(7, 10, 9), we need to find the direction vector of the line.

Let's denote the direction vector as D = (a, b, c). The line can be represented by the parametric equations:

x = 1 + at,

y = 10 + bt,

z = 3 + ct,

To find the direction vector, we can use two points on the line, for example, M and N. The direction vector can be obtained by subtracting the coordinates of M from the coordinates of N:

D = N - M = (4 - 1, 10 - 10, 6 - 3) = (3, 0, 3).

So, the equation of the line passing through M(1, 10, 3) and having the direction vector D = (3, 0, 3) is:

x = 1 + 3t,

y = 10 + 0t = 10,

z = 3 + 3t.

Therefore, the equation of the line is:

x = 1 + 3t,

y = 10,

z = 3 + 3t.

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